3 12 12 triangle

Acute isosceles triangle.

Sides: a = 3   b = 12   c = 12

Area: T = 17.85988213497
Perimeter: p = 27
Semiperimeter: s = 13.5

Angle ∠ A = α = 14.36215115629° = 14°21'41″ = 0.25106556623 rad
Angle ∠ B = β = 82.81992442185° = 82°49'9″ = 1.44554684956 rad
Angle ∠ C = γ = 82.81992442185° = 82°49'9″ = 1.44554684956 rad

Height: ha = 11.90658808998
Height: hb = 2.97664702249
Height: hc = 2.97664702249

Median: ma = 11.90658808998
Median: mb = 6.36439610307
Median: mc = 6.36439610307

Inradius: r = 1.32328756555
Circumradius: R = 6.04774315681

Vertex coordinates: A[12; 0] B[0; 0] C[0.375; 2.97664702249]
Centroid: CG[4.125; 0.99221567416]
Coordinates of the circumscribed circle: U[6; 0.7565928946]
Coordinates of the inscribed circle: I[1.5; 1.32328756555]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 165.6388488437° = 165°38'19″ = 0.25106556623 rad
∠ B' = β' = 97.18107557815° = 97°10'51″ = 1.44554684956 rad
∠ C' = γ' = 97.18107557815° = 97°10'51″ = 1.44554684956 rad

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How did we calculate this triangle?

Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 3 ; ; b = 12 ; ; c = 12 ; ;

1. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 3+12+12 = 27 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 27 }{ 2 } = 13.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.5 * (13.5-3)(13.5-12)(13.5-12) } ; ; T = sqrt{ 318.94 } = 17.86 ; ;

4. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 17.86 }{ 3 } = 11.91 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 17.86 }{ 12 } = 2.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 17.86 }{ 12 } = 2.98 ; ;

5. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 3**2-12**2-12**2 }{ 2 * 12 * 12 } ) = 14° 21'41" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12**2-3**2-12**2 }{ 2 * 3 * 12 } ) = 82° 49'9" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12**2-3**2-12**2 }{ 2 * 12 * 3 } ) = 82° 49'9" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 17.86 }{ 13.5 } = 1.32 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3 }{ 2 * sin 14° 21'41" } = 6.05 ; ;




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